In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. In this paper, a class of weakly inhomogeneous random walks termed Random Walk with Random Transition Probabilities is investigated. As an application, a criterion for the recurrence or transience of these processes in the discrete Abelian case is given. This criterion is deduced using Fourier analysis of Markov additive processes and a perturbation argument of a Markov operator. The latter extends the results of the literature since it does not involve a quasi-compacity condition on the operator. Finally, this criterion is applied to some well known examples of random walks on directed graphs embedded in two dimensional integer lattice. Despite the type problem has been already solved for these examples, one claim this analysis brought a new insight to this problematic.